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But one knows since **the** introduction of powerdomains by Plotkin in [Plo76] that denotational seman- tics can be extended with a reasonable amount of non-determinism, corresponding for instance to a non- deterministic choice operator – non-deterministic extensions of **the** **lambda**-**calculus** **and** of PCF have been designed, with this kind of operational features, **and** powerdomain-based denotational semantics. Even more drastically, if one renounces to **the** domain-theoretic viewpoint on semantics, or more precisely, to **the** fact that **the** domain interpreting **the** types should have some kind of built-in coherence, or compatibility notion, then there are no obstacles to define models of **lambda**-calculi, or of linear logic, which admit non-determinism under **the** guise of **the** possibility of defining arbitrary joins (or unions, or sums) of points.

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Ces idées ont été implémentées dans Dedukti, permettant d’augmenter grandement sa fiabilité. Mots clés : théorie de la preuve, théorie des types, méthodes formelles
Typechecking in **the** λΠ-**Calculus** Modulo: Theory **and** Practice
Abstract: Automatic proof checking is about using a computer to check **the** validity of proofs of math- ematical statements. Since this verification is purely computational, it offers a high degree of confidence. Therefore, it is particularly useful for checking that a critical software, i.e., a software that when malfunc- tioning may result in death or serious injury to people, loss or severe damage to equipment or environmental harm, corresponds to its specification. Dedukti is such a proof-checker. It implements a type system, **the**

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When encoding **the** π -**calculus** in linear logic, π -**calculus** reduction is simulated by shallow proof net reduction (also known as surface reduction in linear λ-calculi [ Simpson 2005 ]). **The** concept of łshallownessž comes from **the** syntax of proof nets, which includes certain constructs called łboxesž, associated with **the** exponential modality !(−) **and** marking sub-proof nets that may be duplicated or erased. Shallow reduction never reduces inside a box. This is reminiscent of weak reduction in **the** λ-**calculus**, which does not reduce under an abstraction. Boxes are therefore a natural way of encoding **the** blocking behavior of prefixes in **the** π -**calculus**, i.e., **the** fact that π .P does not reduce even if P may reduce. Recall however that a box is associated with **the** modality !(−); so, if one kind of prefix (input or output) is associated with **the** presence of a box, then it will be associated with !(−), which implies that **the** dual kind of prefix will be associated with ?(−), which does not come with a box. Therefore, matching input/output duality with linear logical duality forces one prefix to be blocking **and** **the** other to be non-blocking; since it makes little sense for input to be non-blocking, this leads straight to **the** asynchronous π -**calculus**.

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Based on these considerations, in this paper we argue that **the** restriction operator of π-**calculus** does not adequately ensure confidentiality. To tackle this problem, we introduce an operator to program ex- plicitly secret communications, called hide. From a programming language point of view, **the** envisaged use of **the** operator is for declaring secret a medium used for local inter-process communication; exam- ples include pipelines, message queues **and** IPC mechanisms of microkernels. **The** operator is static: that is, we assume that **the** scope of hidden channels can not be extruded. **The** motivation is that all processes using a private channel shall be included in **the** scope of its hide declaration; processes outside **the** scope represent another location, **and** must not interfere with **the** protocol. Since **the** hide cannot extrude **the** scope of secret channels, we can use it to directly build specifications that preserves forward secrecy. In contrast, we regard **the** restriction operator of **the** π-**calculus**, new, as useful to create a new channel for message passing with scope extrusion, **and** which does not provide secrecy guarantees.

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Dynamic compartments may change their nesting structure dynamically, by applying operations for compartment creation, removal **and** merging. These op- erations may influence **the** speed of diverse reactions within compartments, in particular when compartment volumes change (global to local interactions). Vice versa, local reactions within a single compartment may effect global numeric attributes such as volume **and** surface (local to global interaction). Various lan- guages for modeling systems with dynamic compartments were proposed for sys- tems biology [18, 14, 21], but none of them can express physical, chemical, **and** compartimental aspects in a uniform manner, while providing efficient stochastic simulation. Spatial languages such as **the** Brane Calculi [2] or BioAmbients [18] fix a particular set of operators on compartments, **and** provide a special pur- pose solution for these operations. **The** π-**calculus** with polyadic synchronization

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::: ; u
n ] is a list of terms. While **the** structure of **the** usual -**calculus** is isomorphic to **the** structure of natural deduction, this new
structure is isomorphic to **the** structure of Gentzen-style sequent **calculus**. To express **the** basis of **the** isomorphism, we consider intuitionistic logic with **the** implication as sole connective. However we do not consider Gentzen's **calculus** LJ, but a **calculus** LJT which leads to restrict **the** notion of cut-free proofs in LJ. We need also to explicitly consider, in a simply typed version of this -**calculus**, a substitution operator **and** a

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1.1. FUNCTIONS **AND** TERMINATION 19
**The** Decision Problem Hilbert had complemented his consistency program with **the** so-called Entscheidungsproblem (literally, **the** Decision Problem), which consisted in finding an algorithmic procedure that, given a first order formula F **and** an (effective) set of axioms, would output True if F is a syntactic consequence of **the** axioms **and** False if not, in **the** case such a procedure existed. Gödel’s theorems were a shock for many mathematicians, philosophers **and** logicians, **and** moreover, they were a strong indication that an algorithm of **the** Entscheidungsproblem did not exist as well, which had not been hitherto suspected. However, Gödel’s results did not straightforwardly give this negative answer, because their proof did not address **the** topic of computation, which was essential to understand what algorithmic procedures are **and** how they behave. Thus, **the** notion of computation, that had actually been overlooked by mathematicians **and** by logicians since **the** introduction of mathematics, came into light **and** caused intense reflection on its nature. Several alternative paradigms were proposed to provide a formal **and** comprehensive definition of computation. In his proof of **the** incompleteness theorems, Gödel had considered some obviously computable functions that are nowadays known as **the** primitive recursive functions. Integrating some remarks from Herbrand, he then defined **the** set of (partial) recursive functions, despite **the** fact he did not believe them to capture all possible computations (see [100], chapter 17). Church, who had introduced **the** λ-**calculus** in 1928, was convinced that a function was effectively computable iff it could be encoded by a λ-term, but many researchers, including Gödel, were skeptical. Finally, Turing defined his celebrated abstract machine [104] model, ever since known as **the** Turing machines. Turing explicitly conceived his machines by emulating (i.e. imitating in an abstract way) **the** human mind, seen as a device having a finite number of possible states **and** a reading/writing head interacting with an infinite tape, that is empty at **the** beginning of **the** execution (except for finitely many symbols). Very roughly, this captured **the** idea that (1) a human mind (or a cluster thereof) can handle only a finite number of data (i.e. what is already written on **the** tape) **and** this, in finitely many ways (captured by a finite transition function) (2) a human being writes/erases one letter after **the** other. Last, **the** assumption that **the** tape is infinite gives rise to **the** possibility to conduct a computation (or a reasoning) without limitation in space or time (just, **the** computation or **the** reasoning must stop at some point), which is what **the** notions of decidability **and** computability are about.

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In order to guarantee determinacy in **the** context of CCS rendez-vous communication, it seems quite natural to restrict **the** **calculus** so that interaction is point-to-point, i.e., it involves exactly one sender **and** one receiver. 1 In a synchronous framework, **the** introduction of signal
based communication offers an opportunity to move from point-to-point to a more general multi-way interaction mechanism with multiple senders **and**/or receivers, while preserving determinacy. In particular, this is **the** approach taken in **the** Esterel **and** SL [8] models. **The** SL model can be regarded as a relaxation of **the** Esterel model where **the** reaction to **the** absence of a signal within an instant can only happen at **the** next instant. This design choice avoids some paradoxical situations **and** simplifies **the** implementation of **the** model. **The** SL model has gradually evolved into a general purpose programming language for concurrent applications **and** has been embedded in various programming environments such as C, Java, Scheme , **and** Caml (see [7, 22, 16]). For instance, **the** Reactive ML language [16] includes a large fragment of **the** Caml language plus primitives to generate signals **and** synchronise on them. We should also mention that related ideas have been developed by Saraswat et al. [21] in **the** area of constraint programming.

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19th April 2007
Abstract
LJQ is a focused sequent **calculus** for intuitionistic logic, with a simple re- striction on **the** first premiss of **the** usual left introduction rule for implication. In a previous paper we discussed its history (going back to about 1950, or be- yond) **and** presented its basic theory **and** some applications; here we discuss in detail its relation to call-by-value reduction in **lambda** **calculus**, establishing a connection between LJQ **and** **the** CBV **calculus** λ C of Moggi. In particular,

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1.2 **Lambda** **Calculus** **and** Operational Completeness In this paper we consider **lambda** **calculus**, a subject created by Church **and** Kleene in **the** 30’s, which enjoys a very rich mathematical theory. It may seem a priori strange to look for operational completeness with such a computation model so close to an assembly language (cf. Krivine’s papers since 1994, e.g., [22]). It turns out that, looking at reductions by groups (with an appropriate but constant length), **and** allowing one step reduction of primitive operations, **lambda** **calculus** simulates ASMs in a very tight way. Formally, our translation of ASMs in **lambda** **calculus** is as follows. Given an ASM, we prove that, for every integer K big enough (**the** least such K depending on **the** ASM), there exists a **lambda** term θ with **the** following property. Let a t

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assumed. However, it would be interesting to know if this restriction can be dropped.
Problems arising from non left-linear rewriting are directly transposed to left-linear conditional rewriting. **The** semi-closure condition is sufficient to avoid this, **and** it seems to provide **the** counterpart of left-linearity for unconditional rewriting. However, two remarks have to be made about this restriction. First, it would be interesting to know if it is a necessary condition **and** besides, to characterize a class of non semi-closed systems that can be translated into equivalent semi-closed ones. Second, semi-closed terminating join systems behave like normal systems. But normal systems can be easily translated into equivalent non-conditional systems. Moreover such a translation preserves good properties such as left-linearity **and** non ambiguity. As many practical uses of rewriting rely on terminating systems, semi-closed join systems may be in practice essentially an intuitive way to design rewrite systems that can be then efficiently implemented by non-conditional rewriting.

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